Remarks concerning the conformal deformation of riemannian structures on compact manifolds

A NNALI DELLA

S ^CUOLA N ^ORMALE S UPERIORE DI P ^ISA Classe di Scienze

N ^EIL S. T ^RUDINGER

Remarks concerning the conformal deformation of riemannian structures on compact manifolds

Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 3

^e

série, tome 22, n

^o

2 (1968), p. 265-274

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REMARKS CONCERNING THE CONFORMAL

DEFORMATION OF RIEMANNIAN

STRUCTURES ON COMPACT MANIFOLDS

NEIL S. TRUDINGER

(*)

§

¹

Introdnction.

Yamabe

[8] ^seeks

^to establish the

following

^result:

THEOREM :

Any compact

^C°° Riemannian

manifold of

dimension ~a &#x3E; 3

can be

deformned conformally

^{to a C°°} Riemannian structure

of

constant scalar curvature.

The author has

recently

noticed that the

proof

ⁱⁿ

[8]

^{appears incom-}

plete, causing

^the

validity

^{of Yamabe’s} theorem to be in doubt. _{The pur-} pose of this _paper,

apart

^from

indicating

this error, ^{is to} establish the re-

sult itself

under, however,

^some restriction on the curvature of the manifold

(Section 3)

and also to prove ^a

related regularity

result for

arbitrary manifolds, naniely

^{that weak solutions of the} deformation

problem

^are

necessarily

smooth ones

(Section 4).

The

partial

^results of Section 3 are sufficient to show

that,

^{in more}

than 3

dimensions,

^a structure of constant

positive

scalar curvature may be

topologically

deformed into one of constant

negative

scalar curvature.

This had

originally

been derived

by

^Aubin

[1]

^{from Yamabe’s} theorem. Ho-

pefully,

^a

proof

^{of Yamabe’s} theorem for

arbitrary compact

manifolds will be found.

We let be au n

dimensional,

C°° Riemannian

manifold, n ;:&#x3E;

^{3 and let} gij denote its fundamental

positive

^{definite tensor.} We consider an

arbitrary

Pervennto alla Rodazione il 28 Nov. 1967.

(*) This research was partially supported by ^a Ford Foundation Po8tdoctoral grant ^at the Conrant Institute of Mathematical Scienoed, ^{New York} University.

conformal deformation of the

writing

^{it as}

where u

&#x3E; 0,

^{E C°°}

(M).

_

Let

-By R

denote the scalar curvatures of the g,j

re8pectively.

^Then

according

^to

[8],

^{we have the}

^equation

where

denotes the

Laplace-Beltrami

ⁱ

operator corresponding

^to

Thus the

problem

^of

conformally deforming

the structure on M to one

of constant scalar curvature is

equivalent

^to

proving

the existence of a con-

stant R and

positive,

^C°°

(M)

function u

satisfying equation (2).

^We take up the

study

^{of this}

equation

^{in the}

following

^sections.

The author is

grateful

^{to R. Gardner} and J. Moser for many useful discussions

concerning

^this

problem.

§

²

Yamabe’s approach.

Yamabe

[8] considers,

instead of

equation (2),

^the

equations

proving

THEOREM 1.

(Yamabe)

For any q

N,

there exists a con.3tant

~~q

^{and a}

positive

^C°°

function

uq

(normalized by f iuq ^Bq d v = ~ ) ( 1 ) satisfying equation (3).

M

(1) We denote the volume eletueut on M by dv ^{and assume}

267

The solutions

Âq,

⁷ ttq ^are characterized

by

^the

variational problem

^of

minimizing

^{the functional}

over the class of’ functions in

W2 (2) satisfying

We denote the above class

by Aq

^{so that}

It is

readily

^seen

that 1

^is

non-increasing in q

and that as q ^--~

~~V, iq

2013~ where

Since the

proof

ⁱⁿ

[8]

of Theorem 1

(called

Theorem B

there)

^{is some-}

what

unnecessarily involved,

^we insert here a

simpler,

^more direct version.

WTe let ,u

&#x3E;

0 denote the

ellipticity

constant of A u so that for all

points

^{of J.1f.}

The

proof

^is

split

^{into two}

stages :

(i)

Existence in WTe choose a

« minimizing

sequence », ie ^{a se-} quence ^E

~4~

^{with as n --~ oo. Since}

^urn)

^{is bounded in}

(m),

(2) The Sobolev spaces for positive integer ^{and are defined} by

and

where a~~ ^{is a} mnMi-index and is to be understood in the seiiee of distri- bution theory.

it is

weakly precompact.

We show that is in fact

precompact

ⁱⁿ

Wl

The

argument

^{is a standard one} in the calculus of variations

(cf. [6]). By

the Sobolev

imbedding theorem,

^a

subsequence

^{of the}

u(nJ, (which

^{we im-}

mediately

^{relabel as 11(n)}

itself)

converges in to a

function uq

ⁱⁿ

A,y.

Hence for

arbitrary

^E

&#x3E;

^{0 and}

sufficiently large

m, n

depending

^{on t}

Thus

for

sufficiently large

m, ^n. It then follows

(by (7))

^{that a«?i)} converges ^to zcq ⁱⁿ

lV21

^{We then}

clearly

^have

F (u,)

^{and hence} Uq ^{is a} solution of the variational

problem.

^{Note that we} may choose the &#x3E; ⁰ so that

a.e. Also the

vanishing

of the first variation

yields

^{the Euler-La-}

grange

equations

^for ’Uq, ^ie

for

all ~

^E

1111 (M)

^The function 1lq ^{is thns a} weak solution of

equation (.3).

(ii)

Sntoothne8s aoid

positivety.

^We

quote

^a result frome

lliptic theory.

LEMMA. Let u be a tweak

(1~1 )), non-negative,

solution in M

oj’

^the

linear

equation

. where

f

^E

Lr (31),

^r

&#x3E; n/2.

^{Then u is}

positive

and bounded and we the estimates

. The lemma may be

proved

^{in various} ways. Yamabe

[8]

^{proves and uses}

similar statements. These are also consequences of Moser’s work

(see [3],

f5],

^or

[7]).

269

To

apply

the lemma to the

present situation,

^{we observe that u sati-}

sfies a linear

equation

^{of form}

(9)

^with

where

Thus zcq is bounded and

positive. Elliptic regularity theory [2]

then guaran- tees that uq ^{is C°°.}

To establish Yamabe’s theorem from Theorem 1 it suffices to show that the functions

uq

^are

equibounded

and hence

equicontinuous.

^Such

is the purpose of the second

part

of Yamabe’s work

([8],

^Theorem

^(‘).

^In

his

proof, however,

^the

inequality (6.2)

appears to be in ^error. The correct form should be the

inequality

which is insufficient for the remainder of the

proof

to go

through.

^{More in-}

tuitively speaking,

^his

proof

of Theorem C cannot be

expected

to work since it does not

distinguish

^{at all two} facts related to the

problem :

(i)

the presence of the term - Ru in the

equation

^and

(ii)

^the

compactness

^{of 11/.}

Disregarding

^these

factors,

^{one would not}

egpect,

uniform convergence of a

subsequence

^of

the Uq

^{but rather}

merely

convergence in for

any 8 &#x3E;

⁰ to the trivial

solution,

^{as is the case with the}

^problem

where 92 is a bounded domain in Euclidean n-space. See

[4].

§

³ Partial

results.

We will show In the next section that a

subsequence

^of

the ug

^con-

verges in ^a certain sense to a smooth solution of

equation (2).

However the convergence is not

strong enough

^to

imply

^the

non-triviality

of the resul-

ting

solution. We demonstrate in this section that for a

large

^{class of ma-}

nifolds,

the convergence is

sufficiently

^{nice to}

guarantee

^a

positive,

^smooth solution of

(2).

THEOREM 2. There exists a

positive

^{constant 8}

(depending

^on

gii, ^R)

8itch that _c, there exists cc

positive,

Coo solution

of equation (2)

^with

l~ ⁼ A. Thus Yamabe’s is true under this

assumption

^on the metric

gij.

PROOF. In the

equation ⁽⁸⁾

^{consider a test function}

The result is

and hence from

(7)

Writing

^{tv =}

(u,)(9+1)11

^{the above}

inequality

^becomes

Let us

suppose A &#x3E;

^{0 and}

apply

^{the Sobolev} and Holder

inequalities.

^We

obtain,

^thus

since

II1tq

is bounded

independently

of q. Hence if for

large enough q

^{and we obtain}

Clearly

^from

(13),

^this

inequality

continues to hold 0.

Now

choose # N -

^{1. Then we obtain}

and

the uq

^are

subsequently equibounded by

^{the Lemma. A}

subsequence

therefore converges, ^{with its}

derivatives, uniformly

^{to a smooth solution of}

(2),

which is also

positive by

the Lemma.

271

Note that we have the

following

^bound

for ),,

and therefore the conclusion of Theorem 2 will hold if

.lt1

cial case we have then

COROLLARY 1.

Any conzpact

C°° Riemannian

1na.nifold

^icith

sealar can. be

defoi-med

^{to a} C°° Riemannian

of

^{constant scalar curvature.}

Corollary

¹ may in fact be derived very

simply

from Theorem 1. For

implies A C

^0. Let the maximum value ot’ say be taken

if

on at a

point

Then since

J(P)(0

^{we must have}

and heuce

and

accordingly

^the uq ^are

equibounded.

Aubin

[1]

has shown that if ’It 2 3 and R is a

positive constant,

^then gij may be

topologically

transformed into a metric gij with scalar curvature R

satisfying

^{0. Hence as a further}

corollary

^{we have}

COROLLARY 2.

If

^it &#x3E;

3,

^{a structure}

of’ positive,

^{constant scalfcr curva-}

tui-e niay ^be

deforiiied topologically

^{into one}

of

^constant

negative

^{8calar curva.}

Thus the

sign of

^{the tias no}

topological significane

ⁱⁿ

than t1fO

4. A

regularity

^Theorem.

We _prove now the

following

^result

concerning

weak solutions of’ equa- tion

(

THEOREM 3. Let it be a

tt’’ (1t1)

^solution

oj

^an

eqttatioit of

^the

(2).

Then 1i E Coo

(1lI ).

("S) ^{The mean scalar} onrvature is the quantity

PROOF. The function u satisfies

for

all ~ E j~~ {M ).

^{We choose}

an appropriate

^test

function ~ similarly

^{to a}

method of J. Serrin

[5].

^{Defiue u = sup}

(u, 0)

^{and for a}

fixed fl &#x3E;

¹

(to

^be

eventually

^{chosen as in Theorem}

²⁾

^ciefine the functions

where

2q = p +

^1.

-

The function G

(1t)

^{is a}

uniformly Lipshitz

continuous function of it and hence

belongs

^{to Likeuise Observe} also that U and F vanish for

negative

^{u and that}

Let us now substitute in

(17)

^{test functions}

where I

^{is an}

arbitrary, non-negative

^C

1 (M )

function. The result

is, using (7),

and hence

273

Using (19)

^{we then obtain}

Let us

take q

^{now to have}

compact support

^{in a} coordinate

patch

^{of M.}

The

integrals

ⁱⁿ

(21)

may then be

replaced by integrals

^{over a}

sphere S

in En of radius R

where q

= r~

(x)

^E

Co (R).

^{We choose R so that}

Then

applying

the Holder and Sobolev

inequalities

^to

(21)

^{we obtain}

and hence

We

choose fl

^{as in Theorem}

2,

^{ie 1}

C ~ ~ (n + 2)/(n

^-

2)

^{so that}

2q

^N.

Hence we may let in

(22)

^{to obtain} the estimate

Let

SR/2

denote the

sphere

concentric to S of radius

R/2

and choose 1] ⁼ 1 on

~2~ ! i ~~ ( c ^2/R

^on

^{SR .}

^{Then we obtain}

Replacing u by - u,

^{we obtain}

(24)

also for the function u and

employing

a

partition

^of

unity clearly provides

^a

global

^estimate

for some r

~&#x3E; ~

where C will also

depend

^on the local

LN

^{norms of ~c. The} boundedness of it and

subsequently

its smoothness are now conseqnences ot’ the Lemma.

Q.

^{F. D.}

For the functions uq defined in Section

2,

^{we have a} uniform estimate for the norms

and hence a

subsequence

converges in ^a weak solution of

(2).

Hence we have

COROLLARY 3. 1’he

functions

v . ⁱⁿ

yV’_~ (ll~ ) J’or

any E

&#x3E;

^{0 to}

a &#x3E; solution

of equation (2)

^{1c,ith A.}

Yamabe’s tlieorein would thus follow it’ the uon

triviality

of the above solution u could be demonstrated.

This,

^of course, ^we have

shown,

^{in the}

preceding

section but

only

^{under a} restriction A ~ ~ for certain

positive

^c.

Added in

proof (May 1968).

Since this paper ^was

written,

’r. Aubin has found a

proof

^{of Yamabe’s theorem}

by using

^a

completely

different variation nal

approach.

REFERENCES

[1]

T. AUBIN. Sur la courbure scalaire des variétés riemanniennes compactes. C. R. Acad. Sci.

Paris, ^Vol. 262, 1966 pp. ^A 130-133.

[2]

^L. BERS, F. JOHN and M. SCHECHTER. Partial differential equations, Part. II. New York, 1964.

[3]

J. MOSER. On Harnack’s theorem for elliptic differential equations, Comm. Pure Appl. Math., Vol. 14, 1961, pp. 577-591.

[4]

^S. I. POHOZAKV. Eigenfunctions of ^the equation ^0394u + 03BBf(u) = 0, Dokl. Akad. Nauk.

SSSR, ^Vol. 165, 1965 pp. 33-36.

[5] J.

SERRIN. Local behaviour of ^solutions of quasilinear equations, Acta Math. Vol. 113, ¹⁹⁶⁵ pp. 219-240.

[6]

S. L. SOBOLEV. Some applications of functional analysis ⁱⁿ mathematical physics, ^Transl.

Math. Monographs, ^Vol. 7, Amer. Math. Soc., Providence, ^R. I., ^1963.

[7]

^N. S. TRUDINGER. On Harnack type inequalities ^{and their} application ^to quasilinear ellip-

tic equations. Comm. Pure Appl. Math., ^Vol. 20, 1967, pp. 721-747.

[8]

H. YAMADE. On ^a deformation of Riemannian structures on compact manifolds. Osaka Math.

J., ^Vol. 12, 1960 pp. 21-37.